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Abstract

In this paper we deal with one kind of second order periodic-integrable boundary value
problem. Using the lemma on bilinear form and Schauder’s fixed point theorem, we give
the existence and uniqueness of solutions for the problem under Lazer type nonresonant
condition.

Keywords:

1 Introduction and main results

In this paper, we consider the solutions to the following periodic-integrable boundary
value problem (for short, PIBVP):

(1.1)

where is a given T-periodic function in , and ; is T-periodic in t.

Throughout this paper, we assume

(A1) there exist two constants m and M such that

for all and ;

(A2) there exists such that

Recently, boundary value problems with integral conditions have been studied extensively
[6-10]. As we all know Lazer type conditions are essential for the existence and uniqueness
of periodic solutions of equations [1-4]. In [5] the existence of periodic solutions has been considered for the following second
order equation:

Motivated by the above works, we will consider periodic-integrable boundary value
problem (1.1). The main result obtained by us is the following theorem.

Theorem 1Assume that (A1) and (A2) are satisfied. Then PIBVP (1.1) has a unique solution.

This paper is organized as follows. Section 2 deals with a linear problem. There,
using the bilinear lemma developed by Lazer, one proves the uniqueness of solutions
for linear equations. In Section 3, applying the result in Section 2 and Schauder’s
fixed point theorem, we complete the proof of Theorem 1.

2 Linear equation

Consider the following linear PIBVP:

(2.1)

here is a given T-periodic function in , and ; is a T-periodic function. Assume that

(L1) there exist two constants m and M such that

for all . Moreover, m and M suit (A2).

Theorem 2Assume that (L1) and (A2) are satisfied, then PIBVP (2.1) has only a trivial solution.

In order to prove Theorem 2, let us give some following concepts.

First, for any interval , define

It is clear that is a linear space with the norm as follows:

Define a bilinear form on as follows:

for any and . Let

where N suits assumption (L1), and , , , and are some constants. Then .

From and , we can obtain that there exist two constants and such that

for all . Then from assumptions (L1) and (A2), we have

for all , and

for all . Thus, is positive definite on and negative definite on . By the lemma in [1], we assert that if for all , then .

For every x on with , we introduce an auxiliary function

The following lemma is very useful in our proofs.

Lemma 1If, are continuous and satisfy (L1) and (A2), then the following two points boundary value problem

(2.2)

has only a trivial solution.

Proof It is clear that 0 is a solution of two points boundary value problem (2.2). If is a solution of problem (2.2), then . For any , we have

by using (2.2). Integrating the first terms by parts, we derive

By assumption (L1), is positive definite on and negative definite on . These show for , that is, for . The proof of Lemma 1 is ended. □

Proof of Theorem 2 It is clear that PIBVP (2.1) has at least one solution, for example, . Assume that PIBVP (2.1) possesses a nontrivial solution . The proof is divided into three parts.

Case 1: . By Lemma 1 ( and ), PIBVP (2.1) has only a trivial solution. This contradicts .

Case 2: . Denote

Take

From , there are at least two points in the set S, which implies that and . By Lemma 1 ( and ) the two points boundary value problem

(2.3)

only has a trivial solution. Hence we obtain , . By the definitions of a and b, one has

From , we get

This contradicts and .

Case 3: . This case is similar to Case 2.

Thus, we complete the proof of Theorem 2. □

Theorem 3If, are continuous and satisfy (L1) and (A2), then the following PIBVP

(2.4)

has a unique solution.

Proof Let and be two linear independent solutions of the following linear equation:

Assume that is a solution of PIBVP (2.1), where and are constants. Then by the boundary value conditions of (2.1),

By Theorem 3, PIBVP (2.1) has only a trivial solution, which shows

(2.5)

Let be a solution of PIBVP (2.4), where is a solution of the equation

From the boundary value conditions, we have

From (2.5) constants , are unique. Thus, PIBVP (2.4) has only one solution. □

3 Nonlinear equations

Let us prove Theorem 1. Rewrite (1.1) as follows:

(3.1)

where

Define

Fix , introduce an auxiliary PIBVP

(3.2)

To prove the main result, we need the following Lemma 2.

Lemma 2Iffsatisfies (A1) and (A2), then for any given, PIBVP (3.2) has only one solution, denoted asand.

Proof From condition (A2), it follows that

By Theorem 3, PIBVP (3.2) has only one solution . If does not hold, there would exist a sequence such that , . Choose a subsequence of , without loss of generality, express as itself, such that the sequences are weakly
convergent in . Denote the limit as . It is obvious that .

Because the set

is bounded convex in , by the Mazur theorem, we have . Hence,

By the Arzela-Ascoli theorem, passing to a subsequence, we may assume that

and in . Thus, and .

By

one has

which implies , for any . Hence, .

From PIBVP (3.2), we obtain

(3.3)

This shows that is a nontrivial solution of the following PIBVP:

(3.4)

On the other hand, by Theorem 2, PIBVP (3.4) has only zero, which leads to a contradiction.
The proof of Lemma 2 is completed. □

Set

Define an operator by . Applying Lemma 2, .

Lemma 3OperatorFis completely continuous on.

Proof We first prove that F is continuous. Given any such that . Put . From the definition

(3.5)

We would prove that in . If not, then there would be a such that

Utilizing Lemma 2 and Arzela-Ascoli theorem, passing to a subsequence, we may assume
that . Similar to the proof of Lemma 2, we have . Then

(3.6)

Moreover,

Hence, from Theorem 2, . This implies F is continuous. By Lemma 2, for any bounded subset , is also bounded. Hence, applying the continuity of F and Arzela-Ascoli theorem, is relatively compact. This shows F is completely continuous on . The proof of Lemma 3 is completed. □

Proof of Theorem 1 By Lemma 2, Lemma 3 and Schauder’s fixed point theorem, F has a fixed point in , that is, PIBVP (1.1) has a solution .

The following is to prove uniqueness. Let and be any two solutions of equation (1.1). Then is a solution of the equation

Employing (A2), we have

Hence by Theorem 3, . The uniqueness is proved. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Each of the authors, HH, FC and YC contributed to each part of this work equally and
read and approved the final version of the manuscript.

Acknowledgements

The authors are grateful to the referees for their useful comments. The research of
F. Cong was partially supported by NSFC Grant (11171350) and Natural Science Foundation
of Jilin Province of China (201115133).